On a Definition of a Variety of Monadic ℓ-Groups

Abstract

In this paper we expand previous results obtained in [2] about the study of categorical equivalence between the category IRL ₀ of integral residuated lattices with bottom, which generalize MV-algebras and a category whose objects are called c-differential residuated lattices. The equivalence is given by a functor ${{\mathsf{K}^\bullet}}$, motivated by an old construction due to J. Kalman, which was studied by Cignoli in [3] in the context of Heyting and Nelson algebras. These results are then specialized to the case of MV-algebras and the corresponding category ${MV^{\bullet}}$ of monadic MV-algebras induced by “Kalman’s functor” ${\mathsf{K}^\bullet}$. Moreover, we extend the construction to ℓ-groups introducing the new category of monadic ℓ-groups together with a functor ${\Gamma ^\sharp}$, that is “parallel” to the well known functor ${\Gamma}$ between ℓ and MV-algebras.